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Proximal analysis and minimization principles. (English) Zbl 0865.49015

Let \(X\) be a Hilbert space, and suppose \(f:X\to\mathbb{R}\cup\{+\infty\}\) is lower semicontinuous. Given a point \(x\) where \(f(x)<+\infty\), the set \(\partial_Pf(x)\) of proximal subgradients of \(f\) at \(x\) consists of all \(z\) in \(X\) for which there exist a constant \(\sigma>0\) and a neighbourhood \(\Omega\) of \(x\) such that \[ f(y)-\langle z y-x\rangle+\sigma|y-x|^2\geq f(x)\quad\forall y\in\Omega. \] The authors present a self-contained proof of the following density property: given any point \(x_0\) where \(f(x_0)<+\infty\) and any \(\varepsilon>0\), there exists a point \(x\) where \(\partial_Pf(x)\neq\emptyset\) and \(|x-x_0|+|f(x)-f(x_0)|<\varepsilon\). Combining this result with certain properties of the quadratic infimal-convolution, or Moreau-Yosida approximation, they produce efficient proofs of two basic results: (i) key properties of proximal subgradients for functions of the form \(f(x)=\inf_{s\in S}|x-s|\), where \(S\) is a given closed subset of \(X\); (ii) Hilbert-space versions of the well-known variational principles of Ch. Stegall [Math. Ann. 236, 171-176 (1978; Zbl 0379.49008)] and J. M. Borwein and D. Preiss [Trans. Am. Math. Soc. 303, 517-527 (1987; Zbl 0632.49008)]. The delicacy of the density property cited above is illustrated with a construction of a continuously differentiable function \(f\) on \(\mathbb{R}\) for which the set where \(\partial_Pf(x)\neq\emptyset\) has measure zero and fails to contain a dense \(G_\delta\), and the same is true of \(\partial_P(-f)(x)\). Although many of the key facts in this paper are already known, this presentation is unique, since it derives them all from the existence of proximal subgradients on a dense set.
Reviewer: P.Loewen (Bath)

MSC:

49J52 Nonsmooth analysis
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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